256 ANALYTICAL MECHANICS
where r is the density of water. Substituting this expression for m in equation (1)
0 (a + «)«=! [(a + «)»«] at
= (a + kt)*^ + Z(a-rkt)*kv at
, dv , 3k * ,~
and it + ^+ktv = g> (2>'
r s k r 3 k
the integral of which is v = e a+kt | Cge a + kt dt + c | •
/. v = e~31°£ <fl + *3 [JV10* (a + *<> & 4- cl
= (a + ta)~3 [V J(a + &03 d« + cl
/ .7^ .T ( v , 3a2^2 , 3afe2^3 , ^4\ , I = (a + ^)-8^ (^ + — — + -y- + — J + cj
= (a + ^)-3f| (4 a3^ + 6 a%2 + 4 akH* + kH*) + c] .
Let # = 0 when t — 0; then c = 0. fi? 4 a3 + 6 a2Jc^ + 4 a/b2^2 + kH*
PROBLEMS.
1. Find the pressure upon the canvas roof of a tent produced by a shower. The following data are given — the raindrops have a velocity of
50 — '- at right angles to the roof; the intensity of the shower is such as sec.
to produce a deposit of 0.2 inch per hour; 1 cubic foot of water weighs 62.5 pounds.
2. Find the pressure on horizontal ground due to the impact of a column of water which falls vertically from a height of 500 feet.
cm
3. Water flowing through a pipe at the rate of 100 — '• is brought to-
sec.
* Equation (2) is of the form ~t- -{- Py = Q, which is the typical linear
equation, with the integral y = e~~*'p<2aT CQe^^dx + cl.